Abstract: A finite transitive permutation group is said to be -transitive if all the nontrivial orbits of a point stabiliser have the same size greater than 1. Examples include the 2-transitive groups, Frobenius groups and several other less obvious ones. We prove that -transitive groups are either affine or almost simple, and classify the latter. One of the main steps in the proof is an arithmetic result on the subdegrees of groups of Lie type in characteristic : with some explicitly listed exceptions, every primitive action of such a group is either 2-transitive, or has a subdegree divisible by .

John BambergAffiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009
Email:
John.Bamberg@uwa.edu.au

Michael GiudiciAffiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009
Email:
Michael.Giudici@uwa.edu.au

Cheryl E. PraegerAffiliation:
Centre for the Mathematics of Symmetry and Computation, School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley, Western Australia 6009 – and – King Abdulaziz University, Jeddah, Saudi Arabia
Email:
Cheryl.Praeger@uwa.edu.au